Subgroups — C₁₂₇×#13538 #44577211

Order 32512.

294 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#1942	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1845	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1380	C₂	1	2	yes	maximal
4	C₁₂₇×#1380	C₂	1	2	yes	maximal
5	C₁₂₇×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
7	C₂×#107×C₁₂₇	—	2	1	no	—
8	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
9	C₂×#107×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×#107×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	#389×C₁₂₇	—	2	1	no	—
13	#384×C₁₂₇	—	2	1	no	—
14	#340×C₁₂₇	—	2	1	no	—
15	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
18	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
19	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
21	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
22	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
23	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₁₂₇	—	2	2	no	—
25	C₂×C₁₆×C₁₂₇	—	2	2	no	—
26	C₂×C₁₆×C₁₂₇	—	2	2	no	—
27	#107×C₁₂₇	—	2	1	no	—
28	#107×C₁₂₇	—	2	2	no	—
29	#107×C₁₂₇	—	2	4	no	—
30	#107×C₁₂₇	—	4	2	no	—
31	#107×C₁₂₇	Q₈	1	2	yes	—
32	C₄×C₈×C₁₂₇	—	2	2	no	—
33	#95×C₁₂₇	—	4	1	no	—
34	C₂²×C₄×C₁₂₇	—	4	1	no	—
35	C₂²×C₄×C₁₂₇	—	4	1	no	—
36	C₂²×C₄×C₁₂₇	—	4	2	no	—
37	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	—	2	1	no	—
41	C₂×C₈×C₁₂₇	—	2	1	no	—
42	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
43	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	#41	1	2	yes	—
45	C₂×C₈×C₁₂₇	#41	1	2	yes	—
46	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
47	C₄⋊C₄×C₁₂₇	—	2	2	no	—
48	C₄⋊C₄×C₁₂₇	—	4	1	no	—
49	C₄⋊C₄×C₁₂₇	—	4	1	no	—
50	C₄⋊C₄×C₁₂₇	—	4	2	no	—
51	C₄⋊C₄×C₁₂₇	—	4	2	no	—
52	C₄⋊C₄×C₁₂₇	—	4	1	no	—
53	C₄⋊C₄×C₁₂₇	—	8	2	no	—
54	C₄⋊C₄×C₁₂₇	—	8	2	no	—
55	C₄⋊C₄×C₁₂₇	—	8	2	no	—
56	C₄²×C₁₂₇	—	2	1	no	—
57	C₄²×C₁₂₇	—	2	1	no	—
58	C₁₆×C₁₂₇	—	4	2	no	—
59	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	—	2	1	no	—
61	C₂×C₄×C₁₂₇	—	2	1	no	—
62	C₂×C₄×C₁₂₇	—	2	1	no	—
63	C₂×C₄×C₁₂₇	—	2	1	no	—
64	C₂×C₄×C₁₂₇	—	4	2	no	—
65	C₂×C₄×C₁₂₇	—	8	1	no	—
66	C₂×C₄×C₁₂₇	—	8	1	no	—
67	C₂×C₄×C₁₂₇	—	8	1	no	—
68	C₂×C₄×C₁₂₇	—	8	2	no	—
69	C₂×C₄×C₁₂₇	—	8	1	no	—
70	C₂×C₄×C₁₂₇	—	8	2	no	—
71	C₂×C₄×C₁₂₇	—	8	2	no	—
72	C₂×C₄×C₁₂₇	—	8	1	no	—
73	C₂×C₄×C₁₂₇	SD₃₂	1	2	yes	—
74	C₂×C₄×C₁₂₇	Q₃₂	1	2	yes	—
75	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	#123	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	#124	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	#137	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	#137	1	1	yes	characteristic
81	C₈×C₁₂₇	—	2	2	no	—
82	C₈×C₁₂₇	#125	1	2	yes	—
83	C₈×C₁₂₇	#126	1	2	yes	—
84	C₂²×C₁₂₇	#474	1	1	yes	characteristic
85	C₂²×C₁₂₇	#480	1	1	yes	characteristic
86	C₂²×C₁₂₇	#486	1	1	yes	characteristic
87	C₂²×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
88	C₂²×C₁₂₇	C₂×Q₃₂	1	1	yes	characteristic
89	C₂²×C₁₂₇	#509	1	1	yes	characteristic
90	C₂²×C₁₂₇	#509	1	1	yes	characteristic
91	C₄×C₁₂₇	—	2	2	no	—
92	C₄×C₁₂₇	—	2	2	no	—
93	C₄×C₁₂₇	—	8	2	no	—
94	C₄×C₁₂₇	—	16	1	no	—
95	C₄×C₁₂₇	—	16	2	no	—
96	C₄×C₁₂₇	#400	1	1	yes	characteristic
97	C₄×C₁₂₇	#475	1	1	yes	characteristic
98	C₄×C₁₂₇	#483	1	1	yes	characteristic
99	C₄×C₁₂₇	#491	1	1	yes	characteristic
100	#13538	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂×C₁₂₇	#2099	1	1	yes	characteristic
102	C₂×C₁₂₇	#2226	1	1	yes	characteristic
103	C₂×C₁₂₇	#2227	1	1	yes	characteristic
104	C₂×C₁₂₇	#2233	1	1	yes	characteristic
105	C₂×C₁₂₇	#2236	1	1	yes	characteristic
106	C₂×C₁₂₇	#2241	1	1	yes	characteristic
107	C₂×C₁₂₇	#2244	1	1	yes	characteristic
108	#1942	C₂×C₁₂₇	1	1	yes	characteristic
109	#1845	C₂×C₁₂₇	1	1	yes	characteristic
110	#1380	C₂×C₁₂₇	1	2	yes	—
111	#1380	C₂×C₁₂₇	1	2	yes	—
112	#1374	C₂×C₁₂₇	1	1	yes	characteristic
113	C₁₂₇	#13538	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
115	C₂×#107	—	2	1	no	—
116	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
117	C₂×#107	C₂²×C₁₂₇	1	1	yes	characteristic
118	C₂×#107	C₂²×C₁₂₇	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
120	#389	—	2	1	no	—
121	#384	—	2	1	no	—
122	#340	—	2	1	no	—
123	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
124	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
125	C₂×C₄⋊C₄	—	2	1	no	—
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	4	1	no	—
129	C₂×C₄⋊C₄	—	4	1	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
132	C₂×C₁₆	—	2	2	no	—
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	#107	—	2	1	no	—
136	#107	—	2	2	no	—
137	#107	—	2	4	no	—
138	#107	—	4	2	no	—
139	#107	Q₈×C₁₂₇	1	2	yes	—
140	C₄×C₈	—	2	2	no	—
141	#95	—	4	1	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	—	4	1	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
146	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
151	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
152	C₂×C₈	#41×C₁₂₇	1	2	yes	—
153	C₂×C₈	#41×C₁₂₇	1	2	yes	—
154	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	4	1	no	—
157	C₄⋊C₄	—	4	1	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	1	no	—
161	C₄⋊C₄	—	8	2	no	—
162	C₄⋊C₄	—	8	2	no	—
163	C₄⋊C₄	—	8	2	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₁₆	—	4	2	no	—
167	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	SD₃₂×C₁₂₇	1	2	yes	—
182	C₂×C₄	Q₃₂×C₁₂₇	1	2	yes	—
183	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
184	C₂×C₄	#123×C₁₂₇	1	1	yes	characteristic
185	C₂×C₄	#124×C₁₂₇	1	1	yes	characteristic
186	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
187	C₂×C₄	#137×C₁₂₇	1	1	yes	characteristic
188	C₂×C₄	#137×C₁₂₇	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	#125×C₁₂₇	1	2	yes	—
191	C₈	#126×C₁₂₇	1	2	yes	—
192	C₂²	#474×C₁₂₇	1	1	yes	characteristic
193	C₂²	#480×C₁₂₇	1	1	yes	characteristic
194	C₂²	#486×C₁₂₇	1	1	yes	characteristic
195	C₂²	C₂×SD₃₂×C₁₂₇	1	1	yes	characteristic
196	C₂²	C₂×Q₃₂×C₁₂₇	1	1	yes	characteristic
197	C₂²	#509×C₁₂₇	1	1	yes	characteristic
198	C₂²	#509×C₁₂₇	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	8	2	no	—
202	C₄	—	16	1	no	—
203	C₄	—	16	2	no	—
204	C₄	#400×C₁₂₇	1	1	yes	characteristic
205	C₄	#475×C₁₂₇	1	1	yes	characteristic
206	C₄	#483×C₁₂₇	1	1	yes	characteristic
207	C₄	#491×C₁₂₇	1	1	yes	characteristic
208	C₂	C₁₂₇×#2099	1	1	yes	characteristic
209	C₂	C₁₂₇×#2226	1	1	yes	characteristic
210	C₂	C₁₂₇×#2227	1	1	yes	characteristic
211	C₂	C₁₂₇×#2233	1	1	yes	characteristic
212	C₂	C₁₂₇×#2236	1	1	yes	characteristic
213	C₂	C₁₂₇×#2241	1	1	yes	characteristic
214	C₂	C₁₂₇×#2244	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
100. #13538	C₁₂₇	direct (N × H)
113. C₁₂₇	#13538	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₂₇×#1942	2
C₁₂₇×#1845	2
C₁₂₇×#1380	2
C₁₂₇×#1374	2
#13538	127

Subgroups — C127×#13538 #44577211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13538 #44577211